1.)If u + v + w + x + y = 15, then what is the maximum value of uvx + uvy + uwx + uwy?
Notice that
uvx+uvy+uwx+uwy=u(x+y)(v+w)
If u,v,w,x,y are allowed to be negative, there is no largest value, since, for example, one could let u,x,y be arbitrarily “large” negative numbers (so u(x+y) is positive) then v+w is a large positive number and the whole product can be as large as you like.
If they are restricted to positive real numbers, then from AM-GM, we have
≥
Using the initial condition, the left side is just 5 , so we have found
u(x+y)(v+w) ≤ =125
The maximum value is 125 , and occurs if, and only if, u=x+y=v+w=5 .
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